Biomedical Engineering Reference
In-Depth Information
in.a.local.sense..he.path.of.a.ray.through.a.refractive.interface.(e.g.,.air.to.water).is.bent.and.appears.to.
follow.the.path.of.least.time,.accounting.for.the.slowing.down.of.the.light.in.the.higher-index.medium.
hese.observed.phenomena.are.ultimately.explained.by.a.wave.theory.of.light..It.turns.out.that.waves.
add.constructively.and.destructively.in.such.a.way.as.to.favor.traveling.through.a.region.immediately.
surrounding.the.least-time.path..An.added.beneit.of.the.wave.theory.is.that.it.also.accurately.predicts.
the.size.of.the.conducting.region.(e.g.,.how.far.light.rays.can.actually.stray.from.straight.lines!).
We.start.by.deining.the.optical.path.distance.(OPD)..his.is.the.distance.a.phase.front.would.have.
traveled.at.the.speed.of.light.during.the.time.it.spent.traveling.through.the.refractive.medium:
=
=
d
x
x
c n
d
dOPD d
c t
c
c
n x
(
) d
x
=
=
.
v
(
/
)
phase
B
(
)
=
∫
OPD A,B
n x
( )d
x
A
Why.does.light.appear.to.preselect.only.the.path.of.least.time.(or.equivalently,.least.OPD).before.it.
heads.out.on.its.journey.from.A.to.B?.To.answer.this,.let.us.imagine.that.light.emitted.from.source.A.
actually.travels.along.
all the possible
.paths.in.space.from.A.to.B..he.light.that.travels.on.the.shortest.
optical.path.from.A.to.B.takes.a.total.time.equal.to.the.minimum.OPD.between.A.and.B.divided.by.the.
speed.of.light..All.other.paths.require.longer.travel.time,.and.these.waves.have.a.phase.further.advanced.
than.the.wave.taking.the.shortest.path.when.they.arrive.at.B..A.cluster.of.paths.around.the.shortest.path,.
difering.by.no.more.than.one-half.wavelength.in.total.optical.path.length,.will.add.constructively.in.
phase,.contributing.to.intensity.at.B..All.the.other.paths.(an.ininite.number.of.them).have.various.path.
lengths,.which,.modulo.wavelength,.span.the.full.range.of.phases;.and.thus,.these.light.waves.tend.to.
cancel.each.other.and.not.contribute.much.intensity.at.B.
Now,.let.us.put.a.screen.with.a.hole.in.it.at.an.intermediate.plane.between.A.and.B..For.simplicity,.we.
now.consider.A.and.B.and.the.surrounding.space.to.be.of.constant.index.of.refraction,.so.the.minimum.
OPD.path.is.a.straight.line.from.A.to.B..For.the.most.part,.unless.the.screen's.hole.is.located.on.the.line.
from.A.to.B,.the.light.from.A.will.not.reach.B..Assuming.the.hole.is.much.larger.than.a.wavelength,.we.
see.that.the.possible.paths.from.A,.going.through.this.hole,.then.on.to.B.have.a.wide.variety.of.OPDs,.
thus.many.of.the.paths.tend.to.cancel.and.contribute.nothing.to.the.detector.at.B..here.is.one.path,.
however,.through.the.hole.and.on.a.straight.line.from.A.to.B,.which.has.a.cluster.of.nearby.paths.around.
it. with. very. similar. path. lengths,. all. within. one-half. wavelength. of. each. other.. he. wavefronts. along.
those.paths.constructively.add.and.light.is.detected.at.B.
A.little.geometry.shows.how.big.this.cluster.of.paths.is.(Figure.1.6)..If.the.straight.line.distance.from.A.
to.B.is.
L
,.then.the.cluster.of.contributing.paths.(OPD.diference.less.than.λ/2).cover.an.area.with.diam-
eter.approximately.
L
λ
..hese.paths.deviate.from.a.straight.line.by.an.angle.up.to.± λ
L
..his.region.
of.allowable.deviation.from.a.straight.line.is.known.as.the.
Fresnel zone
.
he.Fresnel.zone.size.is.signiicant.at.laboratory.scales.with.visible. light..For.example,.with.visible.
λ.=.0.5.μm.light.and.a.2.m.propagation.distance,.the.allowed.deviation.is.~1.mm.or.0.25.mrad.angle..
his.means.that.if.the.edge.of.an.opaque.screen.is.blocking.the.path.from.A.to.B.but.is.within.½.mm.of.
Intermediate
wavefront plane
Emitter
Detector
L
FIGuRE 1.6
he.light.paths.from.an.emitter.to.a.detector.
