Biomedical Engineering Reference
In-Depth Information
We. may. calculate. the. optical. ield.
U
2
(
x,y
). in. the. focal. plane. at. the. distance.
f
. from. the. lens. using.
the.Fresnel.difraction.integral.(3.2).and.Equation.3.5..If.we.neglect.pre-multiplying.factors,.this.leads.
directly.to.the.amplitude.PSF.given.by
∞
∞
jk
f
d
x y
∫
∫
h x y
(
,
)
P x y
(
,
)exp
(
x x
y y
)
=
+
.
1
.
(3.6)
2
2
1
1
1
2
1
2
1
−∞
−∞
( )
u
ρ .to.model.defocus.and.obtain.the.full.three-.
dimensional.amplitude.PSF.of.the.lens.(Wilson.and.Sheppard.1984;.Booth.2001):
We.may.incorporate.an.additional.phase.factor.
1 2
2
2
π
1
1
2
ρ ρ θ
∫
0
.
h u
( ,
ν φ
, )
=
P
( , )exp
ρ φ
ju
ρ
2
+
j
νρ
cos(
φ θ
−
)
d d
.
(3.7)
0
where.we.have.introduced.the.normalized.optical.coordinates
2
π
λ
r a
f
2
ν
=
kr
sin
α
≈
;
r
2
=
x
2
+
y
2
;
N.A.
=
n
sin
α
.
.
(3.8)
2
2
2
2
and
k za
f
δ
α
2
k z
u
≈
≈
4
δ
sin
2
.
.
(3.9)
2
2
where.
δz
.denotes.the.distance.from.the.focal.plane,.and.α.is.the.semi.angle.of.the.lens.aperture.as.deined.
for.the.numerical.aperture.N.A..=.
nsin
α..he.coordinate.ρ=
r a
2
.(
a
.is.the.physical.radius.of.the.lens).has.
been.introduced.to.normalize.the.radial.range.of.integration.
he.intensity.in.the.focal.region.is.described.by.the.intensity.PSF,.which.is.the.square.of.the..amplitude.
PSF..Examples.for.ideal.and.aberrated.intensity.PSFs.were.calculated.using.Equation.3.7.and.the.cor-
responding.pupil.functions.and.are.shown.in.Figure.4.3.
3.4 effective PSF and image Formation in a confocal Microscope
In.the.preceding.section,.we.developed.an.expression.for.the.amplitude.PSF.of.a.single.lens..But.how.
does.this.relate.to.the.confocal.image?.In.a.confocal.system,.we.will.have.an.illumination.lens.with.the.
amplitude. PSF.
h
1
(
u,v
,φ). and. the. detection. system,. is. described. by. the. amplitude. PSF.
h
2
(
u,v
,φ).. If. the.
object.function.(which.may.represent.the.transmittance,.relectance,.or.dye.concentration).is.denoted.by.
t
,.the.coherent.image.formation.process,.using.an.ininitely.small.pinhole,.yields.the.image.for.a.confo-
cal.setup.(Wilson.and.Sheppard.1984):
.
I
=
| (
h h
)
⊗
t
|
2
.
(3.10)
1 2
Here.⊗.is.the.convolution.operator.and.we.see.that.the.image.is.the.absolute.square.of.the.convolution.of.
the.object.function.
t
.with.an.efective.PSF.of.
h
ef
.=.
h
1
h
2
.of.the.system..In.relection.mode,.the.illumina-
tion.and.the.collecting.lens.will.be.the.same,.hence.we.get
.
2
.
(3.11)
|
|
I
reflection
=
h
1
2
⊗
t
