Biomedical Engineering Reference
In-Depth Information
Aberrated
beam
Detector
Pinhole
Φ
(
r
,
θ
)
FIGuRE 10.5
Simple.sensorless.adaptive.system.consists.of.an.aberrated.input.beam,.a.focusing.lens,.and.a.pin-
hole.detector.
he.intensity.at.the.pinhole.plane.is.given.by.the.modulus.squared.of.the.Fourier.transform.of.the.
pupil.ield:
2
1
[
]
.
∫∫
.
I
( , )
ν ξ =
π
P r
( , )exp
θ
ι ν
r
cos(
θ− ξ
) d d
r r
θ
(10.7)
where.ι.is.the.imaginary.unit,.(ν,ξ).are.the.polar.coordinates.in.the.pinhole.plane,.and.
P
(
r
,θ).is.the.pupil.
function..It.is.assumed.that.the.pupil.is.circular,.with.unit.radius.so.that.
P
(
r
,θ).=.0.for.
r
.>.1..Assuming.
that.the.intensity.of.the.input.beam.is.uniform,.we.deine.
P
(
r
,θ).=.exp[ιΦ(
r
,θ)]..In.the.limit.of.a.point-like.
pinhole,.the.metric.(the.signal.measured.by.the.photodetector).is.proportional.to.the.on-axis.intensity,.
which.is.found.by.setting.
ν
.=.0.in.Equation.10.7..he.metric.takes.a.relatively.simple.form:
2
1
2
π
1
[
]
∫
∫
.
M
=
π
exp
ιΦ θ
( , ) d d
r
r r
θ
.
(10.8)
r
θ
If.the.aberration.amplitude.is.small,.the.exponent.can.be.expanded.to.give
2
1
2
π
1
1
2
∫
∫
.
M
=
π
1
+ ιΦ θ − Φ θ +…
( , )
r
( , )
r
2
r r
d d
θ
.
(10.9)
r
θ
Using.the.aberration.expansion.of.Equation.10.1.in.Equation.10.9.yields
2
= +
ι
π
1
2
∑
∫∫
∑
∑
∫∫
.
M
1
a
X r r
d d
θ−
a a
X X r r
d d
θ+…
.
(10.10)
i
i
i
j
i
j
π
i
i
j
where.we.have.omitted.the.explicit.dependence.of.
X
i
.on.(
r
,θ).for.notational.brevity..he.second.term.on.
the.right-hand.side.of.the.equation.can.be.set.to.zero.if.we.assume.that.the.aberration.modes.
X
i
.have.
zero.mean.value..his.assumption.is.reasonable.in.practice.as.it.is.equivalent.to.choosing.modes.that.do.
not.contain.any.piston.component..As.piston.has.no.efect.on.the.metric.measurement,.this.assumption.
does.not.restrict.us.in.any.practical.way..he.metric.can.then.be.written.as
∑
∑
.
M
≈ −
1
α
a a
j
.
(10.11)
i j
,
i
i
j
where.we.deine
1
∫∫
α =
π
X X r r
d d
θ
(10.12)
.
.
i j
,
i
j
