
238 OPTICS FOR ENGINEERS
Surface A
Surface A
Surface A
0
P
P
z =0
r΄
r
n
Figure 8.5 Geometry for Green’s theorem. The surface consists of the combination of two
surfaces, A and A
0
. The goal is to find the field at the point, P. On the right, the specific case of
light propagating through an aperture is considered. The known field is on the plane z = 0.
the left side of Figure 8.5. Starting with the eld, E, known on the surface A, the goal is to nd
the eld at the point P.
We use a spherical wave, known to be a solution of the Helmholtz equation as an auxiliary
eld,
V
(
x, y, z, t
)
=
e
j
(
ωt+kr
)
r
, (8.7)
where r is the distance from the point P. We employ the positiv