Biomedical Engineering Reference
In-Depth Information
Substituting.this.into.the.Helmholtz.equation,.we.derive.the.relation
2
=
ω
.
k E
2
( )
k
E
( )
k
c
from. which. we. can. conclude. a. linear. free-space. dispersion. relation.
k
.=. ω/c.. he. waveform,. a. sinu-
soid.with.frequency.ω.and.wavelength.λ,.remains.intact.as.it.propagates.through.space.with.time..An.
arbitrary.solution.to.the.Helmholtz.equation.can.be.composed.of.linearly.superposed.sinusoidal.plane.
waves.since.they.each.solve.the.equation.and.the.equation.is.linear..he.linearly.superposed.waves.can.
have.diferent.wavelengths.and.directions..hus,.any.waveform.traveling.in.a.given.direction.propagates.
intact.at.the.speed.of.light,.and.multiple.waveforms.traveling.in.diferent.directions.travel.through.each.
other.without.interaction.
1.3 optical Paths through Materials and Free Space
1.3.1 Dispersion and refraction
In.a.propagation.medium,.such.as.glass,.the.light.wave's.resonant.interaction.with.the.atoms.in.the.glass.
causes.the.phase.front.to.efectively.slow.down..he.slow-down.factor.is.known.as.the.index.of.refrac-
tion,.
n
,.which.is.diferent.for.diferent.materials.and.generally.depends.on.the.frequency.of.the.light..In.
a.refracting.medium,.a.plane.wave.of.a.given.frequency.travels.at.a.phase.velocity,.
v
p
,.given.by
c
.
v
=
p
n
( )
υ
Since.each.frequency.ν.travels.at.a.diferent.speed,.the.medium.is.dispersive,.meaning.that.waveforms.
composed.of.multiple.frequencies.will.change.in.form.as.they.travel.through.the.medium..here.is.no.
dispersion.when.the.light.travels.in.a.vacuum,.where.the.phase.speed.is.
c
,.hence.
n
.=.1.for.all.frequencies.
in.vacuum.
We.can.get.an.idea.of.the.practical.consequences.of.dispersion.by.comparing.refractive.index.for.vari-
ous.materials..
Figure.1.2
.
plots.index.versus.wavelength.for.some.glasses,.crystals,.and.water..All.these.
materials.have.“normal”.dispersion,.where.the.index.gets.smaller.at.the.longer.wavelengths..hat.means.
the.waves.at.longer.wavelengths.travel.faster.than.those.at.shorter.wavelengths..“Anomalous”.dispersion.
is.just.the.opposite:.shorter.wavelength.waves.travel.faster.than.the.longer.wavelength.waves,.at.least.for.
part.of.the.range.of.wavelengths.
he.normal.dispersion.of.glass.accounts.for.refraction.and.the.separation.of.colors.seen.in.a.prism.
(
Figure.1.3
)..Light.waves.that.enter.the.glass.at.a.direction.vector.of.normal.to.the.surface.slow.down.on.
their.leading.edge,.pulling.the.propagation.direction,.what.we.might.call.the.light.ray,.closer.to.normal,.
resulting.in.Snell's.law:
sin
sin
θ
θ
=
n
n
.
1
2
1
2
where.θ
1
.is.the.angle.from.normal.of.the.incident.wave,.θ
2
.is.the.angle.of.normal.of.the.refracted.wave,.
n
1
.
is.the.index.(=.1.in.the.case.of.free.space).of.the.initial.medium,.and.
n
2
.is.the.index.of.the.inal.medium..
Since.blue.light.(short.wavelength).slows.down.more.than.red.(long.wavelength),.the.blue.rays.delect.
the.most.when.passing.through.a.prism,.creating.a.spectrum.on.exit.
Another.consequence.of.dispersion.is.that.a.localized.wave.packet.will.travel.at.a.diferent.velocity.
than.the.phase.velocity,.generally.slower.in.normally.dispersive.materials.and.faster.in.anomalously.
dispersive. materials,. but. never. more. than. the. vacuum. speed. of. light,.
c
.. First,. let. us. deine. what. we.
mean.by.a.wave.packet..We.consider.a.waveform.that.is,.initially,.localized.in.a.region.Δ
x
,.as.shown.in.
