Biomedical Engineering Reference
In-Depth Information
To simplify the notation, one can reason in one dimension without loss of generality. The
spatial pulse of light represented in
Figure 17.1C
is defined as follows:
A
ð
x
Þ
5
A
o
rect
ð
x
Þ
(17.4)
where:
8
<
:
1
2
1
j
x
j
,
rect
ð
x
Þ
5
1
2
1
2
(17.5)
j
x
j
5
0
elsewhere
The FT of
A
(
x
) is equal to sinc(
x
). Therefore, the FT of the light intensity [
A
(
x
)]
2
is
[sinc(
x
)]
2
. To the order 0, it is necessary to add the shifted orders
6
1. The function
A
(
x
6
Δ
x
) can be represented through the convolution relationship:
ð
1
N
x
0
ÞUδð
d
x
0
A
ð
x
6
Δ
x
Þ
5
A
ð
x
6
Δ
x
ÞU
(17.6)
2
N
where
x
0
5
x
6
Δx
. The FT of the function
A
(
x
6
Δx
) will be:
e
i
½
2
πf
x
ð
7
Δx=
2
Þx
½
A
ð
x
6
Δ
x
Þ
5
A
ð
f
x
ÞU
FT
(17.7)
where
f
x
is the spatial frequency. The real part of
Eq. (17.7)
is:
x
5
A
f
x
6
Δ
x
Re FT
A
½
ð
x
6
Δ
x
ð
f
x
Þ
cos
2
π
(17.8)
2
By taking the FT of
Eq. (17.8)
, one can return to
Eq. (17.6)
. Then the object will emit wave
trains with a bandwidth that is influenced by the size of the emitting object. Since
L
wt
is
very small, the spread of wave numbers of monochromatic waves must be large. Hence,
there is a quite different scenario with respect to the classical optics context, in which the
length
L
wt
is large when compared to the wavelength of light.
17.2 Gratings Illuminated by Evanescent Waves
Gratings play a fundamental role in the proposed FTH technique. In view of this, the
present section will introduce arguments originally developed by Toraldo di Francia
concerning evanescent solutions of the grating diffraction equations
[9]
.
The equations of the plane-wave solution for the scalar form of the Maxwell equations must
be considered. The general solution derived by Toraldo di Francia
[9]
is also utilized in this
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