Biomedical Engineering Reference
In-Depth Information
6.2 Wavefront Aberration Simulation for a Spherical object
As.we.have.said,.the.wavefront.in.the.pupil.plane.of.the.lens.can.be.described.by.a.complex.pupil.func-
tion.(Wilson.and.Sheppard.1984)
P r
( , )
A r
( , )exp[ (
i
( , ))]
r
θ
=
θ
ψ
+
ψ θ
(6.1)
.
0
.
where. ψ
0
. is. an. arbitrary. ofset. of. the. phase. and. ψ(
r
,θ). the. change. in. phase. induced. by. the. specimen..
In the. ideal,. unaberrated. case,.
P
(
r
,θ). would. be. constant.. Our. model. specimen. shows. a. variation. in.
refractive.index,.but.no.absorption..We.assume.uniform.illumination.of.the.pupil,.such.that.the.ampli-
tude.
A
(
r
,θ).is.unity,.while.the.phase.ψ(
r
,θ).varies..he.specimen.is.approximated.by.a.spherical.region,.
which. has. an. absolute. diference. of. ∆
n
= − . in. refractive. index. between. the. spherical. region. (
n
1
).
and.the.homogenous.embedding.medium.(
n
0
)..It.is.assumed.that.the.small.change.in.refractive.index.
does. not. cause. a. deviation. of. the. direction. of. the. traced. ray.. A. schematic. diagram. of. the. simulation.
model.is.shown.in.Figure.6.1..Here.γ.represents.the.half.angle.of.the.cone.of.marginal.rays.deined.by.
the.numerical.aperture.(NA),.
n
.sin(α),.of.the.objective.lens..If.the.virtual.specimen.were.immersed.in.
a.substance.of.refractive.index.
n
n
n
1
1
0
0
= ,.for.which.the.lens.has.been.designed,.then.γ = ..Otherwise,.if.
the.specimen.were.immersed.in.a.medium.of.refractive.index,.
n
n
0
= ′
,
γ .would.be.given.by.Snell's.law:.
n
(
( )
)
arcsin sin
n n
.. he. ray. being. traced. is. deined. by. the. unit. vector.
t
. and. the. point.
P
. through.
which.it.passes;.β.is.the.inclination.angle.of.the.ray.to.the.optical.axis.(see.Figure.6.1)..he.functions.
P
(
r
,θ). and. ψ(
r
,θ). are. deined. over. a. normalized. pupil. of. radius.
r
max
.=. 1.. Assuming. the. objective. lens.
obeys. Abbe's. sine. condition. (Born. and. Wolf. 1983),. the. ray. is. mapped. to. the. radial. coordinate. of. the.
pupil. by.
r
=
γ
=
α
′
( )
( )
sin
β
/
sin
γ
.. If. the. length. of. the. ray. section. within. the. sphere. is. denoted. by.
a
(
r
,θ),. the.
phase.function.is.given.by
2
2
π
λ
π
λ
.
ψ θ
( , )
r
=
∆
na r
( , )
θ
=
∆
na
( ,
t m
.
)
(6.2)
where.λ.is.the.wavelength.and.the.coordinates.(
r
,θ).in.the.pupil.plane.may.be.expressed.in.terms.of.the.
vectors.
m
.and.
t
.as.deined.in.
Figure.6.2
..he.vector.
m
.points.from.the.focus.
P
.to.the.center.
C
.of.the.
sphere.and.
t
.is.a.unit-length.direction.vector.with.the.components.
Pupil function
P
(
r
,
θ
)
in the pupil plane
of the lens
θ
r
Objective lens
β
γ
Beam focused to the
bottom of the sample
Embedding medium
of refractive index
n
0
Single ray that is traced
t
Spherical region of
radius
R
and
refractive index
n
1
C
Microscope slide
Scan
P
FIGuRE 6.1
On.the.model.for.the.calculation.of.the.aberration.caused.by.a.spherical.object.
