Biomedical Engineering Reference
In-Depth Information
5.3 experimental Setup
In.the.standard.epi-coniguration.that.is.common.in.luorescence.microscopy,.the.lateral.displacement.
cannot.be.inferred.from.the.wavefront.originating.from.the.focal.spot.because.of.the.reciprocal.geom-
etry..herefore,.to.measure.the.distortion.introduced.by.a.number.of.specimens,.the.single.pass.trans-
mission.geometry.shown.on.the.right.of
.
Figure.5.1
.
as.a.part.of.the.interferometer.described.in.section.
Section.4.6.was.used.to.measure.the.wavefront.in.the.pupil.plane.of.the.lens..Subsequently,.a.Zernike.
mode.analysis.as.discussed.in.Section.4.3.was.performed.on.the.measured.wavefronts.to.obtain.the.coef-
icients.of.the.Zernike.mode.orders.2,.3,.and.4..We.explain.further.how.these.Zernike.coeicients.relate.
to.the.geometric.distortions.we.intend.to.quantify.
5.4 relation between Zernike Aberration
Modes and Geometric Distortion
It.is.assumed.that.the.phase.Ψ(ξ,η).of.the.aberration.function.in.the.pupil.plane.of.the.lens.was.measured.
during.the.previously.discussed.interferometric.experiment..he.Cartesian.coordinates.(ξ,η).refer.to.the.
pupil.plane.of.the.lens,.and.the.radius.of.the.circular.pupil.is.normalized.to.unity..he.wavefront.aber-
rations.can.then.be.described.by.the.pupil.function
.
P
(
ξ η
,
)
=
exp
[
j
ψ ξ η
(
,
)]
.
(5.1)
where.
j
= −1..Here.the.amplitude.term.has.been.dropped.because.amplitude.is.assumed.to.be.constant.
across.the.pupil..he.phase.function.can.also.be.represented.by.a.series.of.Zernike.polynomials:
N
∑
M Z
.
Ψ
( ,
ξ η
)
=
( ,
ξ η
)
.
(5.2)
i
i
i
=
1
where.
Z
i
.is.the.Zernike.polynomial.with.index.
i
,.and.
M
i
,.the.corresponding.mode.amplitude..Generally,.
this.method.allows.one.to.extract.the.mode.coeicients,.including.the.higher-order.terms..However,.the.
information.related.to.the.geometric.distortion.is.contained.within.the.three.lower-order.Zernike.terms.
tip,.tilt,.and.defocus.corresponding.to.the.Zernike.coeicients.
M
2
,.
M
3
,.and.
M
4
..To.relate.the.measured.
Zernike. coeicients. to. displacements. in. the. focal. region,. we. look. at. the. intensity. distribution. at. the.
focus.(Wilson.and.Sheppard.1984):
2
j
u
d
ξ η
2
π
1
∫
∫
( ,
, )
( ,
)exp
(
)
(
η
)
.
I t w u
=
P
ξ η
ξ
2
+
η
2
−
j
ξ
t
+
w
.
(5.3)
2
0
0
Here.we.have.used.the.normalized.optical.coordinates
8
π
λ
α
.
u
=
nz
sin
2
.
(5.4)
2
in.the.axial.direction.and
2
π
2
π
t
xn
sin ;
w
yn
sin
.
.
=
α
=
α
(5.5)
λ
λ
in. the. lateral. direction.. We. note. that. the. polar. coordinates. (ρ,θ). are. related. to. ξ,. η. by. ρ
2
=
ξ
2
+ ,.
η
2
ξ ρ
=
cos
,.and.η ρ
θ
=
sin
..he.term.
n
.sin.α.refers.to.the.NA,.and.λ.is.the.wavelength..he.variables.
x
,.
y
,.
θ
