Biomedical Engineering Reference
In-Depth Information
TABLE 9.1
Coherence Lengths and FTL Pulse Duration of Gaussian Pulses, for Different Pulse Bandwidths
Bandwidth (nm)
10
12
15
20
30
40
60
80
100
Coherence length (μm)
28
24
19
14
9
7
5
4
3
FTL pulse duration (fs)
94
78
63
47
31
23
16
12
9
corresponds to the distance a wave travels in a time interval corresponding to the coherence time. A
light source having a Gaussian spectrum centered at wavelength λ
0
and a bandwidth Δλ has a coher-
ence length given by
2
ln
2
λ
∆
2
L
C
=
⋅
.
(9.1)
π
As an indicator, helium−neon lasers will have high temporal coherence (a few tens of centimeters),
while femtosecond laser will have much smaller temporal coherence (a few tens of micrometers at
most). Coherence lengths of ultrafast lasers, for several pulse bandwidth, can be found in Table 9.1.
Interestingly, Fourier-transform-limited (FTL) ultrafast lasers have a coherence length that actually
corresponds to the spatial extension of the pulse, given by the product of its duration with the speed
of light. The corollary is that non-FTL ultrafast lasers have a coherence length that is smaller than the
spatial extension of their pulses.
The spatial coherence is a measure of how predictable, on average, is the phase of a wave at differ-
ent spatial coordinates. More precisely, the spatial coherence is the time-independent cross-correlation
between two points in a wave. Lasers all have relatively good spatial coherence properties resulting from
stimulated emission of light. In opposition, incandescent light bulbs have very low spatial coherence
properties. A high spatial coherence source can however be made from an incandescent light bulb by
diffraction through a very small aperture, as in Young's double-slit experiment of 1803.
Because holography is based on interferometric principles, a light source having relatively good
coherence properties is needed to record holograms. Similarly, the polarization state of the light source,
and how predictable it is, on average, also impact the recording of a hologram.
9.2.3 Hologram Recording
In holography, the phase and amplitude of the light diffracted by an object is encoded in an intensity-
only image, called hologram, by means of interference with a reference wave (Figure 9.1). Using stan-
dard nomenclature, the light diffracted by the object is referred to as the object wave and is designated
by
o
. Similarly, the reference wave is designated by
r
and is generally a plane or spherical wave. Of
course, both
o
and
r
are complex functions of space and time, but only their time-independent behavior
(wavefront) is of interest here. Indeed, the oscillation frequency of visible electromagnetic waves reaches
almost to the petaHertz range, that is, orders of magnitude faster than the integration time of any detec-
tor. Therefore, only the time-averaged wavefront can be detected. Mathematically, the intensity
I
result-
ing from interference of
o
with
r
at the detector plane can be expressed by
*
1
g
o
r
o r
,
I
=
(
o
r
)
,
(9.2)
g
1
o r
,
where the star symbol (*) denotes the complex conjugate. For interference to occur,
o
and
r
must have
some mutual (temporal, spatial, and polarization) coherence properties, expressed here by the normal-
ized mutual coherence function
g
o,r
.
If the object and reference waves have no mutual coherence (
g
o,r
= 0), then
I
= |
o
|
2
+ |
r
|
2
is simply the
sum of the respective intensity of
o
and
r
. In such case, no holographic information exists. Oppositely, if
