Biomedical Engineering Reference
In-Depth Information
A simple one-dimensional model can be developed to illustrate the abovementioned point.
By utilizing the scalar theory of light wave propagation, one can start from the Fourier
solution of the Maxwell equations:
ð
1
N
1
2
e
i
kx
i
ωð
k
Þτ
d
k
E
ð
x
; τÞ
5
p
A
ð
k
Þ
(17.1)
π
2
N
where
E
(
x
,
) is the scalar representation of the propagating electromagnetic field,
x
is the
direction of propagation of the field,
τ
τ
is the time, A(
k
) is the amplitude of the field,
k
is
the wave number 2
(
k
) is the angular frequency. A(
k
) provides the linear
superposition of the different waves that propagate and can be expressed as:
π
/
λ
, and
ω
p
2
A
ð
k
Þ
5
π
δð
k
2
k
o
Þ
(17.2)
where
(
k
2
k
o
) is the Dirac's delta function. This amplitude corresponds to a
monochromatic wave, i.e.,
E
δ
; τÞ
5
e
i
kx
2
i
ωðkÞτ
. At time
τ
5
0,
E
(
x
,0) represents
(
Figure 17.1C
) a finite wave-train of length
L
wt
where A(
k
) is not a delta function but a
function that spreads a certain length
ð
x
k
. The dimension of
L
wt
depends on the object size
which, in the present case, is smaller than the wavelength of the light.
Δ
In Ref.
[11]
, it is stated that if
L
wt
and
Δk
are defined as the RMS deviations from the
average values of
L
wt
and
2
and
2
, then:
Δk
defined in terms of the intensities
jE
(
x
,0)
j
jA
(
k
)
j
1
2
L
wt
Δ
k
$
(17.3)
Since
L
wt
is very small, the spread of wave numbers of monochromatic waves must be
large. There is a quite different scenario from the classical context in which the length
L
wt
is large compared to the wavelength of light. This simple model indicates that the observed
objects can be considered as electromagnetic oscillators emitting a spectrum of frequencies
that will play an important role in the developments that follow.
A
(
k
)
E
(
x
,
τ
)
L
(
x
)
Δ
k
L
wt
x
x
k
k
0
L
wt
Figure 17.1
(A) Harmonic wave-train of finite extent L
wt
; (B) corresponding Fourier spectrum in wave numbers
k; (C) representation of a spatial pulse of light whose amplitude is described by the rect(x)
function
[10]
.
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