194
Chapter 3. Wave optics
Given this information, we wish to evaluate the amplitude u(x) at
a remote observation point at position x. This point is designated
by the point P in the figure. This represents the statement of our
problem in mathematical terms.
Following Sommerfeld [85], we postulate two point sources at P
and Q, on opposite sides of the screen, and equidistant from it. We
imagine a spherical wave emanating separately from each of the
two points P and Q, where the two waves are assumed to radiate
exactly 180 degrees out of phase relative to one another. The resultant amplitude G is found by algebraically adding the complex
amplitudes corresponding to the two spherical waves (3.235). This
yields
1
1
G = exp(ikR) −
exp(ikR 1 ),
(3.236)
R
R 1
where the radii R and R 1 are shown in the figure. In the plane of
the screen, R = R 1 , and consequently, G = 0. This will be crucially important in the following.
Because G is a superposition of two spherical waves, it is immediately evident that G satisfies the homogeneous Helmholtz equation
v
2 G + k
2 G = 0
(3.237)
everywhere, except at the source points P and Q, where G has
singularities (3.236). We assume here that the differentiation is
with respect to the components of x, shown in Figure 3.9.
We can now write
u(x) v
2 G(x, x 0 ) − G(x, x 0 ) v
2 u(x) dτ
τ
∂
∂
=
u(x)
G(x, x 0 ) − G(x, x 0 )
u(x) dS, (3.238)
S
∂n
∂n
where the left side is an integral over an enclosed volume τ , and
the right side is an integral over the surface S enclosing the volume
τ . We have made direct use of Green’s theorem (3.233), where we
Chapter 3. Wave optics
Given this information, we wish to evaluate the amplitude u(x) at
a remote observation point at position x. This point is designated
by the point P in the figure. This represents the statement of our
problem in mathematical terms.
Following Sommerfeld [85], we postulate two point sources at P
and Q, on opposite sides of the screen, and equidistant from it. We
imagine a spherical wave emanating separately from each of the
two points P and Q, where the two waves are assumed to radiate
exactly 180 degrees out of phase relative to one another. The resultant amplitude G is found by algebraically adding the complex
amplitudes corresponding to the two spherical waves (3.235). This
yields
1
1
G = exp(ikR) −
exp(ikR 1 ),
(3.236)
R
R 1
where the radii R and R 1 are shown in the figure. In the plane of
the screen, R = R 1 , and consequently, G = 0. This will be crucially important in the following.
Because G is a superposition of two spherical waves, it is immediately evident that G satisfies the homogeneous Helmholtz equation
v
2 G + k
2 G = 0
(3.237)
everywhere, except at the source points P and Q, where G has
singularities (3.236). We assume here that the differentiation is
with respect to the components of x, shown in Figure 3.9.
We can now write
u(x) v
2 G(x, x 0 ) − G(x, x 0 ) v
2 u(x) dτ
τ
∂
∂
=
u(x)
G(x, x 0 ) − G(x, x 0 )
u(x) dS, (3.238)
S
∂n
∂n
where the left side is an integral over an enclosed volume τ , and
the right side is an integral over the surface S enclosing the volume
τ . We have made direct use of Green’s theorem (3.233), where we