Biomedical Engineering Reference
In-Depth Information
Zernike.polynomials.are.oten.used.for.systems.with.circular.apertures.as.they.form.a.complete,.orthog-
onal.set.of.functions.deined.over.a.unit.circle.(Zernike.1934;.Born.and.Wolf.1983)..he.expansion.may.
also. be. based. on. the. deformation. modes. of. a. DM. or. the. statistics. of. the. induced. aberrations.. For. an.
eicient.sensorless.AO.scheme,.the.modal.expansion.is.chosen.to.ensure.that.the.metric.function.has.
certain.mathematical.properties..hese.are.discussed.in.Section.10.4.
10.4 Sensorless Adaptive optics Using Modal Wavefront Sensing
In.the.design.of.an.eicient.sensorless.AO.scheme,.we.should.use.a.priori.knowledge.of.the.form.of.the.
optimization.metric.
M
.as.a.function.of.the.mode.coeicients.
a
i
..In.most.practical.situations,.this.form.is.
approximately.parabolic.in.the.vicinity.of.the.maximum.of.
M
..his.observation.assists.us.in.designing.
an.eicient.optimization.algorithm.that.uses.minimal.function.evaluations.and.hence.fewest.specimen.
exposures..We.will.irst.show.the.principle.of.this.approach.in.a.system.where.only.one.aberration.mode.
is.present..In.the.subsequent.discussion,.we.extend.this.description.to.the.case.of.multiple.aberration.
modes.
10.5 Measurement of a Single Mode
When. only. one. aberration. mode.
X
i
. is. present. in. the. system,. then. the. aberrated. wavefront. can. be.
expressed.as.
(
)
(
)
r aX r
i
,.and.the.optimization.problem.is.equivalent.to.a.one-dimensional.maxi-
mization.of.a.parabolic.function..Close.to.its.maximum,.the.metric.
M
.can.be.expressed.as
Φ θ
, =
,
θ
M M
≈
−α
a
.
2
(10.2)
.
0
where.
a
. denotes. the. aberration. mode. coeicient. and.α. is. a. constant.. he. metric. has. the. maximum.
value.
M
0
. when. the. aberration. coeicient.
a
. is. zero.. he. values. of.
M
0
. and.α. are. usually. unknown. as.
they.depend.on.factors.such.as.specimen.structure,.illumination.intensity,.and.detection.eiciency..
Let.us.now.include.the.efect.of.an.adaptive.element.that.adds.the.aberration.
cX
i
(
r
,θ)..Equation.10.2.
then.becomes
M M
≈
−α
(
a c
+
)
2
(10.3)
.
.
0
where.it.is.clear.that.
M
.is.maximum.when.
c
.=.−
a
,.and.the.correction.aberration.fully.compensates.the.
system.aberration..As.we.are.free.to.choose.the.value.of.
c
,.the.right-hand.side.of.Equation.10.2.con-
tains.three.unknown.variables,.one.of.which,.
a
,.we.wish.to.determine..It.follows.that.we.can.extract.
the.value.of.
a
.from.three.measurements.of.
M
.that.we.obtain.using.diferent.values.of.
c
..To.achieve.this.
practically,.bias.aberrations.corresponding.to.three.diferent.values.of.
c
.are.intentionally.introduced.
to.the.wavefront.and.the.corresponding.metric.values.are.measured..his.process.proceeds.as.follows..
he. irst. step. requires. taking. an. image. with. zero. additional. aberration. applied. by. the. adaptive. ele-
ment.(
c
.=.0)..he.metric.value.
M
Z
.is.calculated.from.the.resulting.image..Subsequently,.two.further.
measurements.are.taken,.with.small.amounts.(
c
.=.±
b
).of.the.bias.aberration.introduced.to.the.system..
he.corresponding.images.provide.the.metric.values.
M
+
.and.
M
−
..We.then.use.a.simple.quadratic.maxi-
mization.algorithm.to.ind.the.aberration.magnitude,.
a
,.from.the.three.metric.measurements.(Press.
et.al..1992):
b M M
M
(
)
−
.
+
−
.
(10.4)
a
= −
2
−
4
M
+
2
M
+
Z
−
From.this,.the.necessary.correction.amplitude.is.obtained.as.
c
.=.−
a
..his.measurement.principle.is.
illustrated.in
.
Figure.10.3
.
