Biomedical Engineering Reference
In-Depth Information
function.in.the.vicinity.of.the.paraboloidal.peak..he.new.modes.are.obtained.as.a.linear.combina-
tion.of.the.basis.modes.by.inding.the.coordinate.system.that.aligns.the.primary.axes.of.the.parabo-
loidal. maximum. with. the. coordinate. axes.. his. approach. has. been. demonstrated. in. structured.
illumination,.two-photon,.and.harmonic.generation.microscopes.(Débarre.et.al..2008,.2009;.Olivier.
et.al..2009)..One.possible.implementation.of.this.empirical.method.is.outlined.in.Section.10.9.
he.analytical.approach.is.applicable.to.a.very.limited.range.of.systems..he.numerical.approach.has.
wider.use.and.could.be.used.with.nonanalytic.basis.functions,.such.as.the.deformation.modes.of.a.DM..
However,.this.requires.an.accurate.numerical.model.of.the.mirror.properties.and.full.speciication.of.
other.system.parameters..he.empirical.approach.is.more.widely.applicable,.as.it.could.be.applied.to.any.
sensorless.system,.as.long.as.the.metric.has.a.paraboloidal.maximum..his.method.neither.requires.full.
speciication.of.the.system.nor.an.accurate.model.of.the.DM.properties.
10.9 An empirical Approach for the Derivation of optimal Modes
It. is. possible. to. obtain. optimal. modes. through. experimental. measurements,. avoiding. the. need. for. a.
mathematical.description.of.the.inner.product..One.method.for.achieving.this.is.presented.in.this.sec-
tion..In.principle,.this.involves.analyzing.a.set.of.measurements.of.the.metric. M .with.diferent.applied.
bias.aberrations..From.these.measurements,.one.can.obtain.the.shape.of.the.metric.function.and.deter-
mine. the. orientation. of. the. paraboloid. encoded. in. the. coeicients.α i,j . of. Equation. 10.5.. In. turn,. this.
enables.the.derivation.of.the.optimal.modes.
We.choose.a.set.of.basis.modes. X i ,.the.deinition.of.which.can.be.arbitrary.at.this.point—typically,.one.
would.choose.either.Zernike.modes.or.a.set.derived.from.the.properties.of.the.adaptive.element,.such.
as.mirror.deformation.modes..We.assume.that.the.modes.have.zero.mean.value.and.are.all.normalized.
to.have.a.root-mean-square.phase.of.1.rad..We.also.assume.that.the.basis.modes.do.not.contain.compo-
nents.of.tip,.tilt,.and.defocus;.as.these.components.correspond.to.three-dimensional.images.shits,.they.
should.be.excluded.from.AO.correction.in.a.three-dimensionally.resolved.microscope.system.
For.any.pair.of.basis.modes.with. i .≠. j ,.the.metric.value.is.measured.for.a.number.of.test.aberrations.
Φ ( i,j ), n ,.which.contain.certain.combinations.of.the.modes. X i .and. X j ..he.mode.combinations.are.chosen.
to.have.a.constant.total.magnitude.γ.so.that.the. n th.test.aberration.is
2
2
π
π
.
.
Φ
= γ
  cos
N n X
 
+
sin
N n X
 
(10.15)
( , ),
i j n
i
j
where. N . is. the. total. number. of. test. aberrations.. he. constant. γ. should. be. chosen. small. enough. such.
that.the.assumption.of.small.aberration.magnitude.is.valid;.a.root-mean-square.phase.value.of. < 0.5.rad.
is.usually.appropriate..In.terms.of.the.shape.of.the.metric.function,.the. N .measurements.represent.the.
height.of.the.function.along.a.circular.path,.a.ixed.distance.away.from.the.peak..his.is.illustrated.in.
Figure.10.6 . .he.collection.of. N .metric.measurements.forms.a.polar.plot,.where.the.measured.metric.val-
ues.are.interpreted.as.radial.coordinates.and.the.numbers. n N .as.angular.coordinates..he.resulting.
elliptical.plot.is.a.“section”.through.the.multidimensional.paraboloidal.peak.of. M .that.shows.the.orien-
tation.of.the.paraboloid.to.the.axes..Several.of.these.plots,.acquired.for.diferent.pairs.of.modes,.provide.
suicient.information.to.reconstruct. M .and.obtain.the.principle.axes,.which.correspond.to.the.optimum.
modes..One.can.obtain.the.coeicients.α i,j .by.itting.a.multidimensional.ellipsoid. α
i j .into.
the.data.points,.where. C .is.a.constant..An.example.of.this.process.was.initially.presented.by.Débarre.
et al..(2008).as.applied.to.an.adaptive.structured.illumination.microscope.
he. selection. of. test. aberrations. is. not. restricted. to. the. pairs. of. modes,. as. explained. in. the. above.
illustration..It.should.be.possible,.for.example,.to.use.random.combinations.of.all.basis.modes.to.pro-
vide.random.samples.of.the.metric.function..Similarly,.one.would.obtain.the.coeicients.α i,j .by.itting.a.
multidimensional.ellipsoid.to.the.test.points.
 
a a
 
=
C
i j
,
i j
,
 
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