Biomedical Engineering Reference
In-Depth Information
each.ray.to.produce.a.plane.wave.output.beam,.but.it.compresses.the.bundle.of.rays..he.tilted.incoming.
wave.has.a.more.advanced.phase.front.on.one.side.of.the.aperture.compared.to.the.other.side..Again,.
the.lenses.act.to.bundle.the.rays,.but.since.the.optical.system.adds.the.same.optical.path.length.to.each.
ray,.the.relative.phase.advance.is.the.same.on.the.output.beam.as.it.was.on.the.input..But.since.the.beam.
diameter.is.compressed,.this.must.efectively.increase.the.tilt.of.the.phase.front.
he. light-gathering. power. is. the. product. of. the. maximum. ield. angle. and. the. maximum. aperture..
his.number.is.preserved.throughout.the.optical.system
.
Ξ =
θ
D
=
′
θ
D
′
he.Lagrange.invariant.is.fundamental.physical.concept.in.that.it.is.a.statement.of.the.conservation.
of.energy..he.Lagrange.invariant.squared,.multiplied.by.the.radiance.of.the.sources.over.the.input.ield,.
is.the.total.light.energy.entering.the.optical.system.per.unit.time..In.an.idealization.where.no.light.is.lost.
within.the.system,.the.amount.of.energy.emerging.at.the.output.must.match.the.total.energy.entering.
1.5.3 Maréchal's criterion
As.mentioned.earlier,.the.surface.of.a.lens.or.mirror.must.be.polished.to.great.precision.when.we.are.
dealing. with. the. visible. spectrum. of. light. (0.4-0.7.μm).. In. the. 1940s,. the. French. optical. researcher.
Andre.Maréchal.developed.a.criterion.for.determining.how.accurately.an.optic.surface.must.be.pol-
ished.to.achieve.the.difraction.limit,.that.is,.the.physical.limit.of.focusing.performance.as.set.by.the.
wave. optics. rather. than. by. imperfections. in. the. lens. or. mirror. surfaces. themselves.. he. basic. idea.
is. that. in. order. for. light. rays. to. focus. into. a. bright,. sharp. spot,. they. must. mostly. arrive. at. the. focus.
in.phase,.that.is,.within.one-half.of.a.wavelength.in.optical.path.of.each.other..Figure.1.18.shows.an.
example. of. how. a. nominally. converging. wavefront. can. depart. from. a. perfect. spherical. one. but. still.
meet.this.criterion.
If.the.wavefront.(iso-phase.surface).is.completely.within.two.“Maréchal.plates,”.hypothetical.confo-
cal.surfaces.within.half.a.wavelength.of.each.other,.then.all.the.rays.are.close.enough.in.phase.to.add.
coherently. at. focus.. If. some. portions. of. the. wavefront. are. outside. this. boundary,. then. the. rays. from.
these.regions.would.tend.to,.in.coherent.addition,.cancel.rays.from.other.regions,.reducing.the.overall.
brightness.at.focus..When.the.brightness.is.reduced.in.this.manner,.the.energy.(number.of.photons).is.
not.“lost”.in.the.cancellation.process,.but.instead.falls.in.a.region.surrounding.the.focal.point,.making.
the.focus.appear.blurred.and.dim.compared.to.a.perfectly.coherent.focus.
1.5.4 Diffraction Limit
he.difraction-limited.focus.is.not.an.ininitesimal.point.but.instead.is.set.to.a.inite.size.by.Maréchal's.
considerations..A.nominal.concentric.spherical.wave.has.a.single.point.center..If.the.spherical.wavefront.
Wavefront surface
Focus
Δ
x
<
λ
/2
FIGuRE 1.18
Maréchal's.criterion.for.light.arriving.in.phase.at.focus.
