Biomedical Engineering Reference
In-Depth Information
exp(
−
jkz
jz
)
exp
jk
z
U x y
(
,
)
=
−
(
x
+
y
)
2
2
2
2
2
λ
2
.
.
(3.4)
∞
∞
+
jk
z
∫
∫
×
U x y
(
,
)exp
(
x x
+
y y
)
d
x y
1
1
1
1
2
1
2
1
1
.
−∞
−∞
Now. we. see. that,. apart. from. multiplying. phase. factors,. the. ield. distribution.
U
2
(
x
2
,
y
2
). is. a. Fourier.
transform.of.the.input.distribution.
U
1
(
x
1
,
y
1
).in.plane.
A
.evaluated.at.spatial.frequencies.
f
x
=
2
λ
.and.
x
z
y
=
2
λ ..he.difraction.integrals.are.useful.in.the.following.discussion.of.the.intensity.distribution.
in.the.focal.region.of.a.lens.
f
y
z
3.3 Amplitude PSF of a thin Lens
We. are. now. concerned. with. the. properties. of. the. focal. spot. produced. by. a. thin. lens. as. sketched. in.
Figure.3.3..he.efect.of.an.ideal.lens.is.to.convert.a.plane.wavefront.into.a.spherical.one,.which.con-
verges. into. a. single. point.. he. complex. amplitude. directly. behind. the. thin
†
. lens. is. given. by.
U
1
(
x,y
)..
If.one.illuminates.the.lens.with.a.uniform.plane.wave,.the.efect.of.the.lens.may.be.described.by.the.
introduction.of.a.phase.factor.(Wilson.and.Sheppard.1984):
.
jk
f
U x y
(
,
)
=
P x y
(
,
)exp
(
x
2
+
y
2
)
.
(3.5)
1
1
1
1
1
1
1
2
where. the. exponential. factor. gives. rise. to. a. spherical. wave. that. converges. into. a. point. at. a. distance.
f
.
behind.the.lens.
It.is.important.to.be.aware.that.the.complex.pupil.function.
P
(
x
1
,
y
1
).of.the.lens.takes.the.inite.size.of.
the.lens.as.well.as.lens.imperfections.and.aberrations.into.account..In.the.next.chapters.aberrations.will.
be.modelled.by.such.pupil.functions.
Optical field
U
2
(
x
2
,
y
2
)
in the focal region?
Plane
wavefront
Focal length
f
Optical field
U
1
(
x
1
,
y
1
)
(behind lens)
FIGuRE 3.3
We. seek. to. calculate. the. optical. ield.
U
2
(
x
2
,
y
2
). in. the. focal. region. from. the. ield.
U
1
(
x
1
,
y
1
). directly.
behind.the.lens.
†.
We.neglect.the.displacement.of.the.rays.while.traversing.the.lens.