Biomedical Engineering Reference
In-Depth Information
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Aberration-free
M
M
M
M
-
M
Z
M
+
c
-
a
-
b
+
b
0
0
0
FIGuRE 10.3
Principle.of.modal.wavefront.sensing.of.a.single.aberration.mode..hree.images.are.acquired.with.
intentionally.applied.bias.aberrations.−
b
,.0,.and.+
b
,.and.the.corresponding.metric.values.
M
-
,.
M
Z
,.and.
M
-
.are.calcu-
lated.from.the.images..he.aberration.magnitude,.
a
,.is.then.derived.from.the.three.metric.values.using.a.quadratic.
maximization.calculation.
10.6 Measurement of Multiple Modes
∑
i
i
i
.and.the.optimi-
zation.problem.is.equivalent.to.a.multidimensional.maximization.of.a.paraboloidal.function..Using.a.
Taylor.expansion.around.the.maximum.point,.where.
a
i
.=.0.for.all.
i
,.the.metric.
M
.can.be.expressed.in.a.
general.form.as
When.multiple.aberration.modes.are.present.in.the.system,.then.
Φ θ =
( , )
r
a X r
( , )
θ
∑
∑
.
M M
≈
−
α
a a
j
.
(10.5)
0
i j
,
i
i
j
where.
M
0
.and.α
i,j
.are.unknown.constants..The.one-dimensional.quadratic.maximization.approach.
of.
Section. 10.5.
is. not. obviously. extendable. to. this. more-complicated. multidimensional. function..
In. particular,. we. see. that. the. coefficients. for. each. mode. do. not. appear. as. separable. terms. in. the.
Taylor.expansion..If.we.consider.the.shape.of.the.metric.function,.this.means.that.the.primary.axes.
of.the.paraboloid.do.not.align.with.the.coordinate.axes.formed.by.the.coefficients.
a
i
..This.is.illus-
trated. in.
Figure. 10.4a
.
using. the. simple. case. where. two. aberration. modes. are. present.. Performing.
an. optimization. in. one. aberration. coefficient. will. find. the. maximum. along. a. section. parallel. to.
the.coordinate.axes..Repetition.of.this.process.for.the.second.aberration.coefficient.leads.to.a.com-
bined.correction.aberration.for.both.modes..However,.this.does.not.find.the.overall.maximum.of.
the.paraboloid.(although.further.repetition.of.the.process.would.move.the.correction.closer.to.the.
optimum.value).
In.practical.terms,.this.efect.means.that.a.standard.set.of.basis.modes,.such.as.the.Zernike.polynomi-
als,.will.not,.in.general,.be.the.best.choice.for.the.control.of.a.sensorless.adaptive.microscope..Indeed,.it.
has.been.found.that.each.diferent.type.of.microscope.would.have.its.own.ideal.set.of.modes,.which.do.
permit.independent.optimization.of.each.mode.using.the.quadratic.optimization.method.
he. nonideal. basis. modes. can. be. converted. by. recasting. Equation. 10.5. through. a. coordinate.
transformation.so.that.the.coordinate.system.is.aligned.with.the.paraboloid..he.metric.can.then.be.
expressed.as
