T E E LAW OF REFLECTION FROM STRIPS
Randy L. Haupt
Dept of Electrical Engineering
United States Air Force Academy 80840
Introduction
The law of reflection, the angle of incidence equals the angle of reflection
(relative to the normal at the surface), assumes that the surface is infinite in extent.
Ileviations from the law of reflection occur when the surface is only a few wavelengths
across and the incident field is not normal to the surface. Previous authors have
pointed out deviations from the law of reflection for Physical Optics (PO) calculations
when the electric field is parallel to the edge of the strip or plate [1,2]. This
presentation shows the cause of the deviation and compares PO and integral equation
(IE) results for bistatic scattering from perfectly conducting strips.
I’liysical Optics Bistatic Scattering
Assume the strip lies in the x-z plane and has a width of 2a along the x-axis.
The incident field is either Epolarized (electric field parallel to the z-axis) or Hpolarized (magnetic field parallel to the z-axis). The Epolarized bistatic radar crosssection (RCS) is given by
wliere
2a - width of strip in wavelengths
do - incident field direction (measured from positive x-axis)
6 - RCS direction
Tlie maximum direction of the RCS is found by taking the derivative of S with respect
to
4, setting it equal to zero, and solving the following transcendental equation for 4:
+
+
2sa(c0sq4~ COS^) cot[2lra(co~q5~ cos#) ] = 1
Using L’Hospital’s rule on the left-hand-side of (2) as 4 approaches
verifies the law of reflection for Epolarization (4,=180°-4).
(2)
4,
produces 1 and
H-polarization shows a significant deviation from the law of reflection for
narrow strips. The H-polarized RCS differs from (1) and is given by
uH = 8sa2 sin2b S2
(3)
The maximum direction of the RCS is found by taking the derivative of S sin4 with
respect to 4, setting it equal to zero and solving for 4. The transcendental equation
for lhis case is given by
2na(cosbo
+ cosd) cot[2na(cos~, + cosd) ] = 1 +
+
cos4 cosq5,
cos2fj
sin2i
Tlie maximum scattered field direction is found using Brent’s method [3].
W€H2776-3/0OWO429 SI .OO 01990 IEEE
429
(4)
Figure 1 is a graph of the H-polarized incident field direction vs. the maumuiii
scattered field direction for various sizes of strips. As the strip gets narrower. the
deviation from the law of reflection becomes greater. A graph of the E-polarized
results would be a straight line from (Oo.Oo) to (90°,900). Why the difference? TII~,
E-polarized case has currents running parallel to the z-axis, so the contribution from
one of these current filaments is omnidirectional with respect to 4. On the other hand.
the H-polarized case has currents running in the x-direction. A projection of one of
these current elements is maximum a t +9O0 and zero at b=Oo or 180° and follows a
cosine pattern in between. Thus, the S term is similar to the array factor in antenna
theory and the sin4 and sin4, terms are like the element patterns [4].
Integral Equation Bistatic Scattering
The above results are confirmed using an integral equation approach and a
moment method solution. Figures 2 and 3 are graphs of the incident field direction vs.
the maximum scattered field direction for various sizes of strips and E- and IIpolarizations, respectively. Figure 2 shows some deviations from the law of reflection,
but for the most part, the law of reflection is approximately verified. Near-grazing
incidence produces an endfire pattern from the strip. A comparison between Figures 2
and 3 show the agreement between the PO and IE approaches are quite good for ISpolarization, especially for the wider strips. Figure 4 shows an example bistatic
pattern from a four wavlength strip when the incident field is H-polarized and arrives
a t an angle of loo. The peak of the scattered radiation is at 180’-4 = 27O, and the
angle predicted by the law of reflection for an infinite surface is 180O-4 =loo.
Conclusions
The PO and IE results indicate that the law of reflection holds well for C
polarized incident fields. However, significant deviations occur when the incident field
is H-polarized. As the strip gets smaller and the angle of incidence approaches grazing,
the more the law of reflection does not apply.
References
1. C. A. Balanis, Advanced Engineering Electromagnetics. New York: John Wiley 85
Sons, 1989.
2. J. S. Asvestas, “Physical optics and the direction of maximization of the far-field
average power,” IEEE Trans. Antennas Propagat., vol. AP-34, no. 12, Dec 1986, pp.
1459-1460.
3. W. H. Press, et.al., Numerical Recipes. New York: Cambridge University Press,
1986.
4. P. E. Mast, technical discussions
330
Figure 1. This graph shows the incident field (abscissa) vs. the direction of the
maximum scattered field (ordinate) calculated using PO for H-polarization. The
1 X strip, *** 2 X strip, ooo 4X strip, and xxx 8X strip.
curves are represented by
+++
incident angle (degrees)
Figure 2. This graph shows the incident field (abscissa) vs. the direction of the
maximum scattered field (ordinate) calculated using IE for Epolarization. The curves
1 X strip, *** 2 X strip, ooo 4 X strip, and xxx 8 X strip.
are represented by
+++
43 1
incident angle (degrees)
Figure 3. This graph shows the incident field (abscissa) vs. the direction of the
maximum scattered field (ordinate) calculated using IE for H-polarization. The curves
are represented by
1 X strip, +++ 2X strip, ooo 4X strip, and xxx 8X strip.
+++
Figure 4. The H-polarized bistatic scattering pattern of a 4X wide strip for an incident
field of loo. The peak scattering occurs at 18O0-+ = 27O.
432