Biomedical Engineering Reference
In-Depth Information
9.4.2.1 off-Axis Holography
The off-axis holography was initially proposed to spatially separate the zero-order and different imaging
terms upon hologram reconstruction in classical holography (Leith and Upatnieks, 1962). It therefore
naturally appeared as a good solution to the same problem in digital holography.
However, there are some fundamental differences between classical and digital holography. Among
other things, the pixels of digital sensors are much larger and fewer than the grains of photographic
plates and consequently impose an upper limit on the range of off-axis angle resulting in interference
fringes that can be appropriately sampled. Assuming square pixels of pitch Δ
x
= Δ
y
, the maximum spa-
tial frequency that can be sampled by the sensor is (2Δ
x
)
−1
pairs per unit distance. For two plane waves
of wavelength λ, this corresponds to an off-axis angle θ
MAX
of
λ
θ
=
sin
−1
.
(9.8)
MAX
2∆
x
Yet, because the object wave generally would have a nonzero frequency bandwidth, unlike the plane
wave of this example, the off-axis angle would need to be somewhat smaller than θ
MAX
.
In off-axis digital holography, there are two ways to eliminate zero-order and twin image terms: one
operates in the spatial domain, the other in the spectral domain. To get a good understanding of these
methods, let us consider the case of an off-axis configuration, where an object wave propagates along the
optical axis
z
and a plane reference wave subtends an off-axis angle θ with that object wave, as illustrated
in Figure 9.6a. A detector, located in the
xy
plane, records the hologram resulting from the interference
pattern of
o
with
r
. The intensity of such hologram is given by Equation 9.2 and consists of two imaging
terms (
g
o,r
or
* and
g
o,r
ro
*) and two zero-order terms (
oo
* and
rr
*).
The first method is to spatially separate all terms by numerical field propagation, with the single
2D Fourier transform implementation. For a given wavelength and pixel pitch, judicious selection
of reconstruction distance and off-axis angle should make possible complete separation of all terms.
Unfortunately, by limiting the usable off-axis angle, the pixel pitch also affects the spatial separation
power of off-axis configuration. For that reason, there might still remain a partial overlap between
imaging zero-order terms, as it is the case in Figures 9.5c.
The alternative method is based on spatial frequency filtering. The 2D Fourier transform of the holo-
gram comprises four terms and resembles the sketch of Figure 9.6b. If
r
is a plane wave, then the spec-
trum associated to
rr
* is a Dirac delta, located at the origin of the (ω
x
,ω
y
) plane. The spectrum associated
with
oo
*, a term expressing the autocorrelation of
o
, is also centered at the origin of the Fourier plane
and, if
o
has a bandwidth
B
, has a bandwidth 2
B
that is twice as large. Finally, because
r
is a plane wave,
the two imaging terms have a bandwidth
B
and are modulated by the carrier spatial frequency of the
interference fringes. The off-axis angle θ and the azimuth angle ϕ, respectively, determine the norm and
the angle of the carrier frequency, in polar coordinates. Mathematically, the Fourier transform of the
hologram is
·
F{ }(
I k k
,
)
=
= ⊗
I k k
o
(
,
)
x
y
x
y
·
·
·
·
o k k
*
(
,
)
+ ⊗
o
r k
*
(
+
k
,
k
+
k
)
x
y
x
0
,
x
y
0
,
y
·
·
+
R
2
δ
(
k k
,
)
+ ⊗
r
o k
*
(
−
k
,
k
−
k
),
(9.9)
x
y
x
0
,
x
y
0
,
y
with
2
π
λ
2
π
λ
k
=
sin cos
θ
φ
;
k
=
sin sin
θ
φ
.
(9.10)
0
,
x
0
,
y
