Biomedical Engineering Reference
In-Depth Information
total.phase.change.between.center.and.the.edges.of.the.aperture.is.larger.than.2π.in.this.example,.phase.
wrapping.is.visible.as.sudden.jumps.in.the.otherwise.continuous.phase.proile.
In.reality,.there.will.be.many.scatterers.in.the.CV,.each.contributing.its.own.backscattered.signal.to.
the.amplitude.and.phase.proile.detected.in.the.BFP.(
Figure.14.5d
)
..As.discussed.for.OCM,.their.con-
tributions.are.summed.coherently..If.the.CV.is.small.compared.to.the.distance.to.the.major.aberrating.
layers,.the.light.from.all.scatterers.will.essentially.acquire.the.same.aberrations.on.the.way.toward.the.
objective..However,.due.to.their.diferent.positions.with.respect.to.the.geometrical.focus,.each.backscat-
tered.wavefront.will.have.a.diferent.piston,.tilt,.and.defocus..hese.phase.changes.with.low.spatial.order.
lead.to.constructive.interference.in.some.positions.in.the.pupil.and.to.destructive.interference.in.others..
So.while.speckle.in.OCM.was.an.efect.that.could.be.observed.in.the.microscope.images.(produced.by.
scanning. the. CV. through. the. sample),. CGWS. observes. speckle. already. in. the. wavefront. emanating.
from.one.single.CV..he.speckle.adds.a.random.phase.term.onto.the.wavefront,.which.prohibits.direct.
use. of. the. measured. wavefront. for. correction. and. which. we. call. speckle. error.. Note. how. the. phase.
(
Figure.14.5d
,.lower.row).still.has.local.gradients.resembling.the.speckle-free.case.(
Figure.14.5c
)
.
Since.the.random.speckle.phase.term.comes.from.the.random.distribution.of.scatterers.in.the.CV,.
a. diferent. realization. of. randomly. distributed. scatterers. will. lead. to. a. diferent. speckle. phase. term..
herefore,.we.can.hope.to.average.away.this.term.if.we.are.able.to.measure.speckled.wavefronts.for.a.
multitude.of.diferent.realizations.of.scatterers..Indeed,.a.mathematical.analysis.shows.that.the.ensem-
ble.average.of.the.wavefront.over.speckle.realizations.is.dominated.by.the.incoherent.superposition.of.
the.signals.from.the.individual.scatterers;.the.coherent.efects.responsible.for.speckle.formation.cancel.
out.*.his.means.that.speckle.error.can.be.decreased.by.averaging.over.diferent.realizations.of.scatter-
ers..For.example,.20.realizations.reduce.the.speckle.error.by.a.factor.of.sqrt(20).=.4.5,.which.was.found.
to.be.a.suitable.compromise.in.our.experiments.
If.the.positions.of.scatterers.change.over.time.and.ergodicity.is.satisied,.diferent.speckle.realizations.
can. be. obtained. just. by. waiting. and. repeatedly. measuring. the. wavefront.. However,. many. biological.
samples.are.too.static.at.the.relevant.scale.for.this.to.be.feasible..In.this.case,.a.lateral.displacement.of.the.
CV.can.provide.a.new.wavefront.measurement.with.diferent.scatterers.and.therefore.diferent.speckle..
Of.course,.the.use.of.measurements.from.diferent.lateral.positions.implies.a.certain.amount.of.lateral.
averaging. of. the. aberrations,. so. a. compromise. between. reduction. of. speckle. noise. and. restriction. of.
lateral.averaging.needs.to.be.found.
A. second. problem. associated. with. the. speckelization. of. individual. wavefronts. is. the. singularities.
in.the.phase..Since.an.interferometric.phase.measurement.can.determine.the.phase.only.up.to.2π.(see.
Figure.14.5
),.phase.unwrapping.is.necessary.to.reconstruct.a.wavefront,.which.can.easily.have.a.peak-
to-valley.diference.of.several.wavelengths..Phase.singularities.in.the.speckled.wavefront.can.occur.in.
points.where.the.amplitude.goes.to.zero..herefore,.the.phase-unwrapping.algorithm.used.needs.to.be.
stable. against. singularities.. In. existing. implementations. of. CGWS,. a. virtual. Shack-Hartmann. sensor.
(vSHS;.see.later).was.used.for.phase.unwrapping,.but.other.singularity.stable.algorithms.(such.as.any.
other.least-squares.reconstructor).could.also.be.used.
In.summary,.ensemble.averaging.of.phase-unwrapped.speckled.wavefronts.allows.the.determination.
of.the.aberrations.relevant.for.adaptive.optics.in.microscopy..However,.it.should.be.stressed.that.this.
statement.relies.heavily.on.the.assumption.of.point-like.scatterers.and.their.random.distribution.in.the.
CV..he.assumption.of.point-like.scatterers.is.supported.by.studies.(Mourant.et.al..1998;.Schmitt.and.
Kumar.1998).which.have.found.that.backscattering.is.dominated.by.particles.in.the.size.range.of.λ/4.
to. λ/2.. If. scatterers. predominantly. had. sizes. above. the. wavelength,. their. backscattered. signal. would.
not.have.a.spherical.wavefront.but.instead.contain.at.least.part.of.the.aberrations.present.in.the.incom-
ing. ield.. Similarly,. if. point-like. scatterers. were. arranged. in. a. planar. structure. instead. of. randomly.
*
.
In.principle,.this.is.only.true.up.to.a.certain.maximum.density.of.scatterers,.above.which.coherent.efects.would.domi-
nate.even.ater.ensemble.averaging..However,.this.critical.density.of.scatterers.is.several.tens.of.orders.of.magnitudes.
above.physically.possible.densities,.so.that.this.regime.is.of.but.academic.interest.
