Biomedical Engineering Reference
In-Depth Information
Object
point
Image
Point spread function
FIGuRE 3.1
he.point.spread.function.is.the.image.of.an.ideal.point.and.describes.the.imaging.properties.of.the.
optical.system.
y
x
(
x
1
,
y
1
)
(
x
2
,
y
2
)
R
y
Field
U
2
(
x
2
,
y
2
)
z
Plane B
x
Field
U
1
(
x
1
,
y
1
)
Plane A
FIGuRE 3.2
he.geometry.considered.in.Equation.3.1..he.electromagnetic.ield.in.plane.
B
.may.be.calculated.if.
information.on.the.ield.in.plane.
A
.is.available.
which.describes.the.electromagnetic.(complex).ield.
U
2
(
x
2
,
y
2
).in.the.plane.
B
.as.a.result.of.the.propagation.
from.the.known.ield.distribution.
U
1
(
x
1
,
y
1
).in.the.plane.
A
.(see.Figure.3.2)..Here.
R
.denotes.the.geometric.
distance. between. two. points. (
x
1
,
y
1
). and. (
x
2
,
y
2
). and.
k
= 2π λ. is. the. wavenumber.. Every. .element. of. the.
wavefront,.
U
1
.is.seen.to.cause.a.spherical.wave.of.a.magnitude.proportional.to.
U
1
,.which.is.the.implemen-
tation.of.Huygen's.principle..Making.further.approximations.taking.into.account.that.
z Max x y
(
)
,.
,
1
1
one.may.derive.the.Fresnel.difraction.integral.(Wilson.and.Sheppard.1984):
∞
∞
exp(
−
jkz
jz
)
−
jk
z
d d
.
∫
∫
U x y
(
,
)
U x y
(
,
)exp
((
x
x
)
(
y
y
) )
x y
=
−
2
+
−
2
.
(3.2)
2
2
2
1
1
1
1
2
1
2
1
1
λ
2
−∞
−∞
In.case.the.maximum.distance.from.the.
z
.axis.within.the.plane.
A
.also.obeys.the.stricter.condition
1
2
z
k x
(
2
+
x
2
)
.
.
(3.3)
max
we.can.make.further.simpliications. to.yield.the.Fraunhofer.approximation.by.neglecting.terms.con-
taining.
x
2
.and.
x
2
:
