Biomedical Engineering Reference
In-Depth Information
wavelength.and.the.amplitude.drops.linearly.with.distance.so.that.the.intensity,.the.squared.amplitude,.
from.each.point.source.obeys.an.inverse.square.law.
We.now.imagine.a.surface.that.divides.the.space.between.emitters.and.detectors..he.unique.(and.
controversial).aspect.of.this.theory.is.that.we.can.replace.the.original.emitters.by.spherical.wave.emitters.
of.appropriate.strength.at.each.point.on.the.surface.and.achieve.the.same.solution.on.the.detector.side.
Huygens. was. able. to. explain. relection. and. refraction. with. this. approach,. and. Fresnel. was. subse-
quently.able.to.use.this.model.to.explain.difractive.phenomena.
he.Huygens-Fresnel.integral.is.given.by
{
}
(
)
exp
ik
ρ
x
− ′
x
i
.
( )
=
(
)
(
)
∫∫
u
u x
′
cos
ϕ
x x
,
′
d
2
x
′
(
)
λ
ρ
x
− ′
x
S
where.
u
. is. any. component. of. the. electromagnetic. wave. (with. the. “rapidly. varying”. term. exp{
i
ω
t
}.
ignored),.
ρ
x
). is. the. distance. from.
x
′. on. surface. S. to.
x
. (a. point. in. the. “detector”. half-space),. and.
(
− ′
,
( )
.is.the.angle.of.the.line.from.
x
′.to.
x
.relative.the.normal.to.S.at.
x
′..he.cosine.term.is.known.as.the.
obliquity factor
..he.Huygens-Fresnel.integral.is.a.diferential.description.of.light.propagation..he.ield.
at.points.
x
′.on.surface.
S
.is.propagated.to.points.
x
.on.another.surface.further.into.the.detector.half-space..
As.such,.the.ield.is.propagated.from.surface.to.surface.in.a.general.direction.from.sources.to.detectors.
he.factors.
i
.and.λ.have.interesting.interpretations..hese.are.normalizing.factors.that.are.necessary.
to,.among.other.things,.preserve.energy.in.the.wave.as.it.propagates..he.
i
.factor.is.known.as.the.Gouy.
phase.shit.and.says.that.the.Huygens.emitter.must.be.out.of.phase.with.the.original.wave.at.that.point.
by.π/2..Imagine.for.the.moment.that.the.surface.
S
.was.completely.an.opaque.screen.except.for.a.tiny.
(wavelength.size).pinhole.at.
x
′;.this.pinhole.seems.to.act.like.it.instantaneously.shits.the.phase.of.the.
impinging.original.wave.as.it.passes.through.the.pinhole..his.strange.but.true.phenomenon.has.actu-
ally.been.observed.in.experiments.(Ruin.et.al..1999)..he.λ.factor.is.also.very.interesting..It.hints.at.the.
size.of.a.photon—from.a.theory.that.predates.the.photon.theory.by.more.than.two.centuries!.As plane.
wave. .electromagnetic. energy. passes. through. our. hypothetical. small. hole. in. surface.
S
,. it. reduces. the.
amplitude.by.the.factor.1/λ.but.difracts.it.to.all.angles.in.an.expanding.spherical.wave..If.the.hole.is.
made.bigger.(
D
.>.λ),.then.the.difraction.is.less,.now.concentrated.over.a.cone.angle.of.λ/
D
,.but.has.total.
transmitted.amplitude.of.
D
2
/λ.(the.hole.area.is.
D
2
,.and.the.Huygens.normalizing.factor.is.1/λ),.resulting.
in.a.forward.amplitude.proportional.to.
D
..If.the.hole.is.smaller.(
D
.<.λ),.then.the.forward.amplitude.is.
proportional.to.
D
2
..hat.is,.the.forward.energy.(amplitude.squared).getting.through.a.pinhole.is.propor-
tional.to.
D
2
.until.the.hole.gets.smaller.than.λ,.at.which.point.the.forward.energy.drops.drastically.with.
hole.size..his.is.exactly.the.behavior.one.would.expect.of.particles.with.size.roughly.λ!
he.Huygens.integral.seems.to.imply.that.each.point.in.free.space.is.a.reemitter.of.waves,.masking.the.
original.emitters..Even.if.this.is.not.really.the.case.physically,.from.a.mathematical.standpoint.we.can.
heuristically. “prove”. the. Huygens. integral. from. Maxwell's. equations. by. realizing. that. the. Helmholtz.
wave.equation.is.linear.in.the.ield.amplitude.and.so.can.be.formed.by.a.linear.superposition.of.basic.
solutions..he.basic.solution.in.this.case.is.the.wave.that.passes.through.a.screen.pinhole..Adding.up.the.
results.of.screens.with.pinholes.(each.of.“size”.λ.and.experiencing.the.Guoy.phase.shit).results.in.the.
superposed.solution.
Substituting.a.plane.wave.for.
u
φ
x x
(
x
.in.the.Huygens-Fresnel.integral.results.in.an.identical.amplitude.
but.appropriately.phase-shited.plane.wave.
u
( )..he.normalizing.factor.
i
/λ.was.actually.discovered.in.
this.manner.(it.is.required.to.preserve.the.amplitude.and.give.the.correct.phase),.but.the.equation.cor-
rectly.maintains.the.planar.iso-phase.surface.solution.
We.now.return.to.the.idea.that.the.Huygens.integral.is.a.diferential.explanation.of.wave.propaga-
tion.and.we.ind.an.interesting.connection.to.the.minimum.time.principle..Imagine.substituting.any.
number. of. intermediate. Huygens-wavelet-generating. surfaces. between. surface. A. and. surface. B.. he.
Huygens. wavelets. propagate. from. one. surface. and. add. up. coherently. at. the. next. surface.. hen. each.
′
