Biomedical Engineering Reference
In-Depth Information
7.2.2 Angular Spectrum Method
Once a hologram is acquired, it is reconstructed by numerically propagating the optical
field along the direction perpendicular to the hologram plane (
z
-direction) in accordance
with the laws of diffraction. A Fresnel
Kirchhoff integral can be expressed as
[4]
:
ZZ
E
0
ðx; y
;
0
A
0
ðk
x
; k
y
;
0
Þ
5
Þ
exp
½
2
i
ðk
x
x
1
k
y
yÞ
d
x
d
y
(7.1)
where
k
x
and
k
y
are spatial frequencies corresponding to
x
and
y
, respectively. Here,
E
0
(
x
,
y
;
z
5
0) is the intensity distribution recorded by the CCD camera. This equation is the
expression for Fourier transform and
A
0
(
k
x
,
k
y
; 0) is the angular spectrum of the optical field
E
0
(
x
,
y
,
z
5
0) at the hologram plane
z
5
0. The object's angular spectrum consists of a zero
order (DC) and a pair of first-order terms: the angular spectrum of the object field and its
phase inverted version.
Figure 7.2A
shows the hologram of a USAF (United States Air
Force) resolution target recorded by our dual-wavelength experimental setup. The two
crossing interference fringe patterns, formed by each of the two wavelengths, can be clearly
seen.
Figure 7.2B
presents the Fourier spectrum with the two pairs of first-order
components, corresponding to the two wavelengths.
Field
E
0
(
x
,
y
;
z
5
0) can be regarded as a projection of many plane waves propagating in
different directions in space and with the complex amplitude of each component equal to
A
0
(
k
x
,
k
y
; 0). The angular spectrum can then be propagated in space along the
z-
axis using
the complex transfer function exp
½
i
k
z
z
:
Aðk
x
; k
y
;
zÞ
5
A
0
ðk
x
; k
y
;
0
Þ
exp
½
i
k
z
z
(7.2)
q
k
2
where
k
z
5
. Note that there is no requirement for
z
to be larger
than a certain minimum value, as in the case of Fresnel transform or Huygens convolution.
The complex field at an arbitrary
z
can be obtained by performing the inverse Fourier
transform:
2
k
x
2
k
y
and
k
5
2
π
/
λ
ZZ
Aðk
x
; k
y
;
zÞ
Eðx; y
;
zÞ
5
exp
½
i
ðk
x
x
1
k
y
yÞ
d
k
x
d
k
y
(7.3)
As both integrals in
Eqs. (7.1)
and
(7.3)
are computed via the FFT (Fast Fourier transform)
algorithm, the angular spectrum method is well suited for the real-time imaging.
Once the complex field is calculated, its phase can be converted to height. If the light wave
reflects from an object, its surface is described by a height map
h
(
x
,
y
), which is determined
from the phase map
ϕ
(
x
,
y
) of the holographic reconstruction at a given wavelength by:
hðx; yÞ
5
λ
r
2
π
ϕðx; yÞ
(7.4)
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