Biomedical Engineering Reference
In-Depth Information
M
d
n
a
m
d
m
a
n
d
m
d
n
(a)
(b)
(c)
FIGuRE 10.4
(a). Principle. of. modal. wavefront. sensing. for. a. system. with. two. basis. modes,.
X
m
. and.
X
n
..
Maximization.with.respect.to.these.modes.is.not.optimal.and.several.cycles.are.required.to.reach.the.peak..(b).he.
equivalent.process.for.the.derived.modes.
Y
m
.and.
Y
n
..As.the.axes.of.the.paraboloid.are.aligned.with.the.coordinate.
axes,.the.maximization.of.each.mode.is.independent.and.the.peak.is.obtained.ater.a.single.measurement.cycle..(c).
Schematic.illustration.of.the.biased.aberration.measurements.for.the.two-mode.system..he.positions.of.the.ive.
dots.represent.the.aberration.biases.required.for.the.measurements..his.set.of.measurements.is.suicient.to.recon-
struct.the.whole.paraboloid.and.hence.to.determine.the.correction.aberration.
∑
.
M M
≈
−
β
d
.
(10.6)
2
0
i
i
i
where.the.variables.
d
i
.are.new.coordinates.derived.from.the.original.coordinates.
a
i
.and.the.constants.
β
i
.are.related.to.the.constants.α
i,j
.of.Equation.10.5..his.coordinate.transformation.is.equivalent.to.
the. derivation. of. a. new. set. of. aberration. modes.
Y
i
(
r
,θ),. each. of. which. is. a. linear. combination. of.
the.original.modes.
X
i
(
r
,θ)..he.total.aberration.can,.therefore,.be.expressed.in.terms.of.either.set.of.
modes.as.
∑
∑
Φ θ =
( , )
r
a X r
( , )
θ =
d Y r
( , )
θ
..Methods.for.derivation.of.the.new.modes.are.discussed.
i
i
i
i
i
i
in
.
Section.10.8.
As.each.aberration.coeicient.
d
i
.appears.in.a.separate.term.on.the.right-hand.side.of.Equation.10.6,.
it.is.possible.to.employ.the.one-dimensional.quadratic.maximization.in.sequence.to.each.separate.coef-
icient..As.these.calculations.are.mutually.independent,.only.one.maximization.per.mode.is.necessary.
to.ind.the.peak.of.the.metric.function..his.is.illustrated.for.the.two-mode.system.in.Figure.10.4b..he.
orientation.of.the.coordinate.axes.with.the.axes.of.the.paraboloid.ensures.that.the.correction.aberration.
is.found.eiciently.with.the.smallest.number.of.maximization.cycles..We.can,.therefore,.consider.the.
choice.of.modes.
Y
i
.as.being.optimum.for.this.system.
he. single-mode. maximization. procedure. presented. in.
Section. 10.5
.
required. three. measurements.
with.bias.aberrations.corresponding.to.
c
.=.−
b
,.0,.
b
..his.procedure.can.be.applied,.in.turn,.to.each.of.
the.multiple.aberration.modes..It.appears,.therefore,.that.maximization.of.
M
.when.
N
.aberration.modes.
are.present.would.require.3
N
.measurements..However,.as.the.bias.aberration.corresponding.to.
c
.=.0.is.
common.to.all.modes,.the.total.number.of.measurements.can.be.reduced.to.2
N
.+.1..his.is.illustrated.
in.Figure.10.4c.
10.7 example of a Sensorless Adaptive System
In. this. section,. we. illustrate. the. principles. of. sensorless. modal. wavefront. measurement. using. a. sim-
ple.adaptive.optical.system,.as.shown.in.
Figure.10.5
.
.he.measurement.system.consists.of.an.adaptive.
element,.a.convex.lens,.a.subwavelength-sized.pinhole,.and.a.photodetector..he.intensity.of.the.light.
passing.through.the.pinhole.is.chosen.as.the.optimization.metric..his.is.an.appropriate.choice.as.the.
measured.intensity.will.be.maximal.for.an.aberration-free.beam..We.derive.analytic.expressions.show-
ing.how.the.metric.depends.on.the.aberrations.in.the.system.and.we.demonstrate.how.choosing.speciic.
properties.of.the.aberration.modes.allow.its.simpliication.to.the.form.given.in.Equation.10.6.