Principles of Wave Optics - Adaptive Optics for Biological Imaging - page 18

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FIGuRE 1.13 Difraction.pattern.of.a.circular.beam.propagated.to.the.far.ield..(Courtesy.of. http://en.wikipedia.
.org/wiki/File:Airy-pattern.svg .)

1.5 Application of Wave optics to optical System Design

Now.with.a.irm.adherence.to.the.theoretical.ideas.just.presented.that.light.waves.add.coherently.and.
that.the.phase.of.the.wave.is.sum.of.the.OPD.along.the.shortest.path.from.source.to.detector,.we.can.
present.some.basically.useful.properties.for.optical.design.application..Lenses.or.powered.mirrors.mod-
ify.the.path.of.the.light.to.bring.it.into.focus..It.is.useful.to.know.just.how.sharp.a.focus.is.physically.
allowable.in.the.wave.theory..his.can.then.be.compared.with.that.achieved.by.a.real.system..In.the.case.
of.a.pristine.free-space.environment,.the.lenses.or.mirrors.must.be.very.precise,.igured.to.a.fraction.of.
a.wavelength.to.achieve.coherent.addition.

1.5.1 ray tracing
Typically,.designers.use.ray.tracing.to.characterize.and.help.tune.an.optical.design..A.ray.enters.a.lens.glass.
at.an.of-normal.incident.angle.and.is.bent.according.to.Snell's.law.of.refraction.as.shown.in. Figure.1.14 ..
his.law.is.actually.a.consequence.of.the.Fermat.principle.and,.more.fundamentally,.Huygens.wavelet.the-
ory..As.a.ray.enters.the.lens,.the.light.is.slowed.down..his.bends.the.ray.since.the.portion.of.the.iso-phase.
wavefront.that.has.not.yet.contacted.the.lens.surface.moves.ahead.of.that.portion.that.is.now.in.the.glass.
Note.that.as.the.speed.of.light.is.reduced.so.is.the.wavelength,.but.the.frequency.remains.unchanged,.
according.to

c c

ν = =

.

n
1 1 2 2
A.focusing.lens.is.thicker.in.the.middle,.which.slows.down.the.center.rays.with.respect.to.the.edge.
rays..Incident.plane.wave.rays.emerge.from.the.lens.with.the.center.rays'.phases,.lagging.the.phases.of.
those.rays.emerging.from.the.lens.edges..A.correctly.igured.lens.will.refract.an.incident.plane.wave.into.
a.converging.spherical.wavefront.( Figure.1.15 ) ..Similarly,.the.correctly.igured.mirror.will.relect.a.plane.
wave.into.a.converging.spherical.wave.front.
he. total. optical. path. length. through. the. center. of. a. focusing. lens. is. longer. than. the. path. going.
through.the.edges..Standing.at.the.focus,.the.bent-edge.ray.has.traveled.a.longer.distance.in.the.air.than.
the.central.ray;.however,.since.the.central.ray.spent.more.time.(more.optical.path).in.the.glass.lens,.the.
two.OPDs.are.actually.equal.as.measured.starting.from.the.incident.plane.wave.
We. can. exploit. this. idea. to. derive. useful. properties. about. the. optical. focus,. including. transverse.
resolution.and.depth.of.ield.
To.continue.further,.it.is.necessary.to.deine.a.few.more.practical.terms.
he. focal length .is.the.distance.from.the.center.of.the.lens.to.the.focus..he. power .of.the.lens,.in. diop-
ters ,.is.the.reciprocal.of.the.focal.length.of.the.lens.in.meters.

n λ λ