Biomedical Engineering Reference
In-Depth Information
contour on a beam plot is called the full-width-half-maximum (FWHM). A curious outcome
of the radiation from these apertures is that there is a region in which the beam narrows.
The depth where the last axial peak occurs is called the transition distance or natural focal
length,
F
N
. This transition depth demarcates two regions, one with peaks and valleys, called
the near field, and one with a beam with a single peak diminishing in amplitude width and
broadening with distance, called the far field, as shown in Figure 16.20. The transition
depth for a circular aperture of radius
a
is
2
z
t
¼
a
=
l
ð
16
:
42a
Þ
For a rectangular aperture, the transition distance for an aperture
L
x
in the
x
-
z
plane is
2
z
t
L
x
=ð
pl
Þ
ð
16
:
42b
Þ
The natural focal length is the distance to the last axial peak and is approximately the tran-
sition distance.
The far-field beam pattern for a rectangular aperture is the Fourier transform of the
amplitude across the aperture. In the case of uniform illumination
Y
Y
A
ð
x
0
,
y
0
,0
Þ¼
ð
x
0
=
L
x
Þ
ð
y
0
=
L
y
Þ
ð
16
:
43a
Þ
where
8
<
9
=
0
j
x
j >
L
=
2
Y
ð
x
=
L
Þ¼
1
=
2
j
x
j¼
L
=
0
ð
16
:
43b
Þ
:
;
1
j
x
j <
L
=
2
the far-field pattern in the
-
plane is a sinc function
x
z
oÞ¼
L
x
p
l
z
¼
L
x
p
l
z
p
e
i
p=4
sin
ðp
x
x
=l
z
Þ
ðp
L
x
x
=l
z
Þ
p
L
x
x
l
z
0
0
e
i
p=4
sinc
p
ð
x
,
z
,
p
p
ð
16
:
44a
Þ
A plot of this pattern is shown in Figure 16.22.
In the case of a uniform amplitude
u
0
on a circular aperture, the far-field pattern is the
two-dimensional Fourier transform of the circularly symmetric aperture function
jinc
2
2
2
l
z
lÞ
ip
p
a
J
ð
2
pr
a
=ðl
z
ÞÞ
0
p
a
r
a
l
z
0
1
p
ðr
,
z
,
¼
ip
ð
16
:
44b
Þ
2
pr
a
=ðl
z
Þ
where
J
1
is the Bessel function of the
fi
rst kind,
(
)
¼
2
J
1
(2
p
x
)/(2
p
x
), and
r
is the radial
jinc
x
distance to an observation point at (
). A plot of this pattern is shown in Figure 16.22.
Note that the shapes of the far-field patterns are maintained with distance as their ampli-
tudes fall and beams broaden with distance.
From these far-field patterns, it is easy to determine the FWHM beam widths. For the
rectangular aperture in the
r
,
z
x
-
z
or
y
-
z
plane
FWHM
¼
1
:
206
l
z
=
L
ð
16
:
45a
Þ
where
L
is the appropriate aperture for that plane. Similarly, for a circular aperture
FWHM
¼
0
:
7047
l
z
=
a
ð
16
:
45b
Þ
